A187198 McKay-Thompson series of class 12H for the Monster group with a(0) = 5.

1, 5, 14, 36, 85, 180, 360, 684, 1246, 2196, 3754, 6264, 10226, 16380, 25804, 40032, 61275, 92628, 138452, 204804, 300040, 435672, 627356, 896400, 1271525, 1791324, 2507426, 3488472, 4825531, 6638688, 9085888, 12373992, 16772908, 22633812

Offset

-1,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Links

G. C. Greubel, Table of n, a(n) for n = -1..2500

J. M. Borwein and P. B. Borwein, A cubic counterpart of Jacobi's identity and the AGM, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701.

D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of (b(q^2) * c(q^2))^3 / (b(q)^2 * c(q) * b(q^4) * c(q^4)^2) in powers of q where b(), c() are cubic AGM theta functions.

Expansion of (1/q) * chi(q) * chi(q^3) * chi(-q^6)^4 / chi(-q)^4 in powers of q where chi() is a Ramanujan theta function.

Expansion of (eta(q^2) * eta(q^6))^6 / (eta(q)^5 * eta(q^3) * eta(q^4) * eta(q^12)^5) in powers of q.

Euler transform of period 12 sequence [ 5, -1, 6, 0, 5, -6, 5, 0, 6, -1, 5, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = f(t) where q = exp(2 Pi i t).

a(n) = A058486(n) = A187091(n) unless n=0.

a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2017

Example

1/q + 5 + 14*q + 36*q^2 + 85*q^3 + 180*q^4 + 360*q^5 + 684*q^6 + 1246*q^7 + ...

Mathematica

QP = QPochhammer; s = (QP[q^2]*QP[q^6])^6/(QP[q]^5*QP[q^3]*QP[q^4]* QP[q^12]^5) + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 16 2015, adapted from PARI *)

Prog

(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^6 + A))^6 / (eta(x + A)^5 * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A)^5), n))}

Crossrefs

Cf. A058486, A187091.

Sequence in context: A261055 A320853 A193557 * A097507 A052951 A048745

Adjacent sequences: A187195 A187196 A187197 * A187199 A187200 A187201

Keyword

nonn

Author

Michael Somos, Mar 06 2011

Status

approved